The category of modules over a curved $A_\infty$ category has a model structure described by Positselski. If the category is a curved dg-algebra, then the homotopy category of modules contains the category of matrix factorizations as a full and faithful subcategory, and is equivalent to it if one allows infinite rank matrix factorizations. We show how, through the use of an enveloping algebra, the homotopy theory of the module categories can be used to produce a homotopy theory for the categories themselves. This homotopy theory is invariant with respect to ``$b$-deformations" and reduces to the usual notion of homotopy equivalence when the category is uncurved.This is a joint work with Patrick Clarke.